Parents don’t have to be christened before they can have their child christened. Visit this page to learn more about the requirements for christenings.
You can have your child baptized at a christening regardless of whether or not you, the parents, are baptized.
It is only the godparents who are required to be baptized, (so the only time a parent would have to be baptized is if they were also acting as a godparent).
In some situations, finding three baptized godparents can be difficult. It’s possible for parents to step in as godparents – this page explains what is required in this case.